The $Z_2$-homology of a surface viewed as a simplicial complex allows us to extract interesting invariants from the resulting homology groups. $\beta_0$ is the number of connected components, $\beta_1$ is the number of "tunnels" and $\beta_2$ is the number of "cavities" ($\beta_i$, the $i^{th}$ Betti number, being the rank of the $i^{th}$ homology group)

Are there other "interesting" invariants of surfaces (or even graphs) that can be obtained by going from $Z_2$ to some other field (like the rationals), or are all of these equivalent in some sense ?

The number of connected components can be detected by the rank of the zeroeth homology group with any coefficient. Once you assume that everything is connected, there is only $H_1$ and $H_2$. In general, the Betti numbers are, by definition, the ranks of the cohomology groups which may or may not be the same as the ranks of the homology groups. Moreover, there is a (classical) complete classification for connected surfaces - the no. of cavities, the genus, and the orientability.
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Somnath BasuMar 30 '12 at 2:51

So you're saying that they are all equivalent ?
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Suresh VenkatMar 30 '12 at 2:10

I am saying that if you know the homology with integral coefficients, then you can determine the homology with coefficients is any other Abelian group. If you know the homology with coefficients in say $\mathbb{Q}$ then you cannot determine the integral homology.
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Liviu NicolaescuMar 30 '12 at 9:45